Amplification factors in shock-turbulence interactions: Effect of shockthicknessDiego A. Donzis

Citation: Physics of Fluids (1994-present) 24, 011705 (2012); doi: 10.1063/1.3676449 View online: http://dx.doi.org/10.1063/1.3676449 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Shock structure in shock-turbulence interactions Phys. Fluids 24, 126101 (2012); 10.1063/1.4772064 ShockTurbulence Interaction AIP Conf. Proc. 1333, 1391 (2011); 10.1063/1.3562837 Direct numerical simulation of canonical shock/turbulence interaction Phys. Fluids 21, 126101 (2009); 10.1063/1.3275856 Modeling the effect of upstream temperature fluctuations on shock/homogeneous turbulenceinteraction Phys. Fluids 21, 025101 (2009); 10.1063/1.3073744 Modeling shock unsteadiness in shock/turbulence interaction Phys. Fluids 15, 2290 (2003); 10.1063/1.1588306

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Amplification factors in shock-turbulence interactions:Effect of shock thickness

Diego A. Donzisa)

Department of Aerospace Engineering, Texas A&M University, Texas 77843-3141, USA

(Received 30 September 2011; accepted 6 December 2011; published online 11 January2012)

Amplification factors of streamwise velocity are investigated in canonical shock-

turbulence interactions. The ratio of laminar shock thickness to the Kolmogorov

length scale is suggested as the appropriate parameter to understand data from sim-

ulations and experiments. The different regimes of the interaction suggested in the

literature can also be understood in terms of this parameter. VC 2012 AmericanInstitute of Physics. [doi:10.1063/1.3676449]

The interaction of turbulence with shock waves is an important phenomenon in a variety of

contexts including supernovae explosions, supersonic aerodynamics and propulsion, among

others. To understand the complexities of such flows, substantial efforts have been devoted to the

simplest case of isotropic turbulence interacting with a normal shock. A widely used measure of

the effect of a shock wave on turbulence is the so-called amplification factors which are defined as

the ratio of suitably defined averages (typically plane averages for simulations or time averages

for experiments) of quantities of interest after and before the shock. Our interest here is restricted

to amplification factors of velocity components normal to the shock, G � u0d2=u0u2 where prime

stands for root-mean-square quantities and subscripts u and d stand for upstream and downstream

of the shock, respectively. Ribner1,2 and Moore3 assumed the shock to be a discontinuity and the

incoming flow to be a superposition of simple waves and obtained analytical expressions for a va-

riety of amplification factors. Since this theory is based on the linearized Euler equations, it is usu-

ally referred to as the Linear Interaction Analysis, or LIA. A number of other theoretical

approaches have been proposed to predict the effect of a shock on velocity, vorticity, or other

quantities.4–10 However, LIA is currently the most widely used approach to try to understand ex-

perimental and numerical data, in part, because of its relative success over other approaches and

its ability to provide analytical results for a range of quantities of interest.

Evidence from experiments and simulations, however, has consistently shown that the inter-

action depends on characteristics of the incoming flow not accounted for in LIA, such as velocity

and length scales.11 These dependencies, though, are not understood even in a general qualitative

manner. It has been further suggested that the interaction can be in either the “wrinkled” or

“broken” regimes12–14 depending on whether the shock preserves its identity as a well defined

steep gradient or its structure is fundamentally modified. The theoretical approaches mentioned

above cannot account for this behavior. In particular, the relation between the regime of the inter-

action and the scaling of amplification factors has not been investigated in any detail. Since the re-

gime is based on details of the structure of the shock, it is natural to inquire about the role of the

shock thickness in the scaling of the interaction.

In view of this situation, it seems desirable to examine the most recent data available on

amplification factors, investigate the extent to which assumptions needed for theories are satisfied,

and establish how G scales with governing parameters, especially the shock thickness.

Elemental dimensional analysis on the problem of a stationary shock interacting with iso-

tropic turbulence yields

G ¼ f ðMt;Rk;DMÞ; (1)

a)Electronic mail: donzis@tamu.edu.

1070-6631/2012/24(1)/011705/6/$30.00 VC 2012 American Institute of Physics24, 011705-1

PHYSICS OF FLUIDS 24, 011705 (2012)

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where Mt �ffiffiffi3p

u0=c is the turbulent Mach number,15 c the mean speed of sound, Rk � u0kq=l is

the Taylor Reynolds number, k is the Taylor microscale, q and l are the mean density and viscos-

ity and f is an unknown universal function. The parameter DM � M � 1 is chosen over M (the

mean Mach number) for convenience.

To meaningfully compare different flows several conditions have to be met. First, flows have

to be geometrically similar. In our case, we focus on a stationary shock interacting with isotropic

turbulence consisting mainly of vortical fluctuations.16 This leaves out some experiments17 where

the shock moves through spatially and temporally decaying turbulence and some data from simu-

lations where entropy or acoustic fluctuations are artificially introduced at the inlet. Second, the

governing parameters should be obtained at the same location. A natural choice is to use values

immediately upstream of the shock wave, a convention used throughout the literature and here-

after (unless noted). This location is easily identified in simulations as the streamwise rms velocity

reaches a well defined minimum. Third, to properly define G, a location downstream of the shock

has to be identified to measure u02d . In simulations it is common to use the location where u02 peaks

after the shock.18 This extremum has been consistently found in simulations, though has not been

observed in experiments, presumably because of the difficulties in measuring close to the shock.

This point is also critical to compare with theories. LIA provides results in the so-called near field,

where quantities undergo variations as a function of the distance from the shock, and the far field

which represent the asymptotic value far away from the shock. Although amplification factors are

typically compared against far-field predictions, due to the viscous decay after the shock in simu-

lations, it is not clear where the asymptotic value can be found. Still, as commonly adopted in

practice, we use the peak mentioned above to compare with LIA (though others14 extrapolate u0 to

the nominal shock location.) Fourth, either Reynolds or Favre averages have to be used to com-

pute G. Given the already scarce data available, it is unfortunate that experimental and numerical

studies have used Reynolds and Favre averages, respectively. The data presented here comes

mostly from direct numerical simulations (DNS) using Favre averages.

The studies we consider are listed in Table I and include shock-resolving13,19 and shock-

capturing14,20–22 simulations as well as an experimental study23 which satisfy as closely as possi-

ble the conditions above (one difference is the use of Reynolds averages). Fig. 1 shows the collec-

tion of amplification factors from these studies as a function of DM. This representation (often

using simply M for the abscissa) is based on LIA which is formally valid only for Mt ! 0 and

Rk !1 in which limit, the dependence on these parameters is assumed to vanish,

G ¼ fLIAðDMÞ: (2)

TABLE I. Sources: For all simulations the inflow is statistically isotropic and is convected into the domain using Taylor’s

hypothesis. All values correspond to conditions right before the shock, except for those in parenthesis which are at the inlet

of the domain. SC and SR stand for shock-capturing (open symbols) and shock-resolving (closed symbols) simulations

respectively.

Source Mt Rk M Method Inflow Symbol

Lee, Lele, and Moin13 0.0567–0.11 12–20 1.05–1.20 SR “Synthetic” with model spectrum,

no thermodynamic fluctuations

"

Hannappel and Friedrich20,a (0.17)b (6.67) (2.0) SC “Synthetic” with model spectrum ^

Lee, Lele, and Moin21 0.09–0.11 15.7–19.7 1.5–3.0 SC Decaying compressible turbulence h

Mahesh, Lele, and Moin22 (0.14) (19.1) (1.3) SC Decaying compressible turbulence D

Jamme et al.19 0.173 5–6 1.20–1.50 SR Decaying compressible turbulence �Larsson and Lele14,c 0.16–0.38 40 1.3–6.0 SC “Blended” decaying compressible 5

Barre, Alem, and Bonnet23 0.011 15 3 Experiment 3

aAmplification factors taken from Jamme et al.19

bNumbers in this reference are slightly different due to different definitions.cAmplifications factors here are different than in this reference since values were extrapolated to the shock location from

far-field conditions. Values presented here correspond to the location of the peak as described in the text and were provided

by the authors.

011705-2 Diego A. Donzis Phys. Fluids 24, 011705 (2012)

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This type of reduction in the number of similarity parameters in specific limits is usually referred

to24 as complete similarity (or similarity of the first kind), in our case, in Rk and Mt.

Clear discrepancies between data and theory are seen which do not appear to decrease as DMincreases. Careful investigation of the data reveals systematic trends at fixed DM. For example,

Reynolds and turbulent Mach numbers effects are observed at DM � 0:5 where the triangles cor-

respond to simulations at Rk � 40 with Mt varying from 0.155 (top) to 0.375 (bottom) while the

closed circle corresponds to Rk � 5:5 and Mt � 0:133. Taken together, the data suggest both a sys-

tematic decrease of G with Mt and an increase with Rk.

Because a number of theoretical approaches, including LIA, assume the shock to be a discon-

tinuity, their validity rests on the comparison of the shock thickness to other relevant scales, in

particular the smallest dynamically active scale in the incoming turbulence—the Kolmogorov

scale g � ð�3=h�iÞ1=4where h�i is the average dissipation rate (angular brackets represent a suita-

ble average). Since, for moderate Mach numbers, the laminar shock thickness scales25 as

dl � ðl=qcÞDM�1, one can easily show that

dl=g � K; (3)

where K � Mt=ðR1=2

k DMÞ for which we have use the well-known relation h�i � �u02=k2. The

precise value this ratio should assume in order to make theories applicable, cannot be obtained

from dimensional considerations alone and should be compared against data.

Furthermore, from a physical perspective one can argue that when turbulence interacts with a

rapid mean deceleration such as a shock, a relevant non-dimensional parameter governing the

interaction would be the ratio of characteristic scales of the turbulence to those of the shock. Such

a similarity parameter may indeed be K.

This parameter also seems to be a natural choice to demarcate the different regimes of the

interaction. Because K is the ratio of the laminar shock thickness to the Kolmogorov length scale,

for relatively small values of K, the shock is expected to be subjected to a locally uniform velocity

field and will therefore behave as a sharp front corresponding to the laminar solution—that is the

“wrinkled” regime. On the other hand, for relatively large K, the turbulence can greatly disturb the

shock, creating multiple compression peaks, or even smooth compressions—that is, the “broken”

regime.

We are thus interested in investigating the scaling,

G ¼ f �ðKÞ: (4)

From a similarity scaling perspective,24,26 it is interesting to highlight the difference between Eqs.

(2) and (4). While the number of similarity parameters in both have been reduced to one, Eq. (2)

represents a complete similarity solution where two parameters are neglected in the limit of being

very large or very small, whereas Eq. (4) represents incomplete similarity (or of the second kind)

in that the number of parameters is reduced by combining powers of the original similarity param-

eters: dependencies on viscosity, turbulence intensity, and length scales are retained.

To test Eq. (4) against the data, in Fig. 2 we show the collection of amplification factors as a

function of K. A more universal behavior emerges as results from different sources and at different

FIG. 1. Amplification factors from references in Table I.

011705-3 Amplification factors in shock-turbulence interactions Phys. Fluids 24, 011705 (2012)

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conditions appear to follow a single curve; unlike Fig. 1 no systematic trends are apparent for

fixed K which supports K as a more appropriate non-dimensional parameter to characterize the

interaction, at least, under the conditions in simulations and experiments. Overall, the amplifica-

tion factor takes larger values at low K (around 1.6) and lower values at high K (close to unity at

the highest K available).

To try to understand the behavior of G we first consider the limit DM!1 (i.e., K ! 0).

Under all the conditions necessary to make LIA a valid approximation, it is possible to obtain ana-

lytically an asymptotic amplification of G � 1:14 for c ¼ 1:4.9 An asymptotic amplification is not

inconsistent with the data in Fig. 2, though the limiting value (horizontal dashed line) appears to

be about 40% larger than LIA prediction. We note, however, that besides K ! 0, LIA requires

simultaneously Rk !1 and Mt ! 0. Due to the significant viscous decay observed in the simula-

tions, though, it is unclear to what extent an inviscid approximation is justified.

At higher values of K, the amplification factor decreases with K suggesting that as the dispar-

ity between turbulence scales and the shock thickness decreases, the overall effect of the shock on

the incoming flow weakens. This is qualitatively consistent with theoretical8,27 and numerical

results14 which suggest that jumps under turbulent conditions are weaker than in laminar flows. In

this range, a power law of the form G � s1K�s2 is found to represent the data well with s1 � 0:75

and s2 � 1=4 (Fig. 2).

In the other limit DM! 0, and under an inviscid assumption, the shock becomes a weak

Mach wave and the velocity jump vanishes, i.e., G! 1. The available data in the literature do not

appear to achieve high enough K to test such an asymptotic state (Fig. 2). However, we are inter-

ested in finite Reynolds numbers where dl is finite and the turbulence undergoes a viscous decay.

If one could let DM ! 0 such that dl remains finite but the effect on turbulence (at least on large

areas across the shock) is negligible, then the amplification factor will simply reflect the decay of

u02 over a distance dl. Using the classical estimates du02=dt � �h�i � �Au03=L and u02L3 ¼ con-

stant (as a result of the constancy of Saffman integral28) and integrating we obtain29

G � ½1þ Cðu0=L0Þt�r (where C ¼ 5A=6, r ¼ �6=5, and A � 0:4). Using Taylor hypothesis

(t ¼ x=U), the decay over dl is G ¼ ½1þ Cðuu=LuÞðdl=UÞ�r ¼ ½1þ CðMt=MÞKR�3=2

k �r which, for

M not too close to unity or fixed M, may be approximated by G � ð1þ CK2=RkÞr, also shown in

Fig. 2 at high K for Rk � 5, 10, and 20 (dashed-dotted lines). This result suggests that at high K a

Rk-dependence reappears. For fixed Reynolds number, the asymptotic state would scale as

G � K2r. It is not clear whether such an asymptotic state with a steady shock is realizable in prac-

tice, though, as this regime may indeed correspond to “destructive interactions”30 in which vigor-

ous turbulence interacts with a very weak shock. Clearly, high-fidelity (shock-resolving) data are

necessary to establish this regime with any certainty.

It is important to recognize the difficulties in DNS due to the conflicting requirements to

resolve broadband turbulence and capture shocks which make numerical details have a significant

effect on the solution.31 For example, at low K, while Ref. 21 used ENO for inviscid fluxes only in

the streamwise direction and close to the shock and sixth-order compact schemes for the other

terms and in other regions, Ref. 14 used WENO in all directions when a sensor detects steep gra-

dients, otherwise, an explicit sixth-order central scheme is used. One wonders if the numerical

details of different simulations (in particular if grid-convergence tests have not been performed14)

FIG. 2 Amplification factors from Fig. 1 according to Eq. (4).

011705-4 Diego A. Donzis Phys. Fluids 24, 011705 (2012)

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contribute to some of the scatter across studies and whether shock-resolving simulations (solid

symbols) would provide more consistent results at lower K. In addition, since one cannot rule out

an influence of the details of the turbulence at the inlet (see Table I) as well as some statistical var-

iability, the good collapse in Fig. 2 appears to be quite satisfactory.

As noted, K may also be indicative of the different regimes in the interaction. Ref. 13 sug-

gested M2t =ðM2 � 1Þ as the parameter controlling the boundary between regimes. Ref. 14, how-

ever, suggests that data does not support this as the appropriate parameter though it provides some

reasonable guidance with a transition around M2t =ðM2 � 1Þ � 0:06 which corresponds to K � 0:1

at the conditions of the simulations. This is in fact close to the midpoint between asymptotic states

in Fig. 2. Thus, the transition between asymptotes in the figure appears to correspond to the transi-

tion between “wrinkled” and “broken” regimes. Data in Ref. 13 are also consistent with this find-

ing: simulations at K � 0:07 (estimated from the conditions stated in that reference) had relatively

well-defined fronts while at K � 0:48 multiple compression waves were seen along individual

streamlines.

We note, however, that the definition of “wrinkled” and “broken” regimes in the literature is

typically based on a qualitative assessment of contour levels or instantaneous profiles of pressure

or density. The transition in Fig. 2, on the other hand, is relatively smooth and no particular value

of K can be chosen unambiguously to delimit “wrinkled” and “broken” regimes. Therefore, using

the value of K to characterize the interaction seems to be more appropriate: lower K would corre-

spond to better defined sharp fronts across the entire shock surface while higher K would corre-

spond to increasingly distorted fronts and where increasingly large areas may present multiple or

smooth compressions.

We have investigated the extent to which K ¼ dl=g can characterize amplification factors

from simulations and experiments at finite Reynolds and Mach numbers. The main result shown

in Fig. 2 shows that K can, at least to first order, collapse available data into a universal curve.

This result could also be useful to design simulations and experiments depending on what regime

is of interest. In particular, shock-resolving simulations at a range of values of K especially in as-

ymptotic regimes will be highly desirable.

Helpful comments and data from J. Larsson, S. Jamme, and Y. Andreonopolus are gratefully

acknowledged.

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15The factorffiffiffi3p

is included for consistency with the widely used definition Mt ¼ffiffiffiffiffiffiffiffiffiffiffiuiuih i

p=c (summation implied).

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011705-5 Amplification factors in shock-turbulence interactions Phys. Fluids 24, 011705 (2012)

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011705-6 Diego A. Donzis Phys. Fluids 24, 011705 (2012)

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